3D Ice Cream

Have you ever had (or at least seen) Neapolitan ice cream? It’s the kind with chocolate, vanilla, and strawberry, usually as separate layers in one package. As a kid, I didn’t care for the strawberry. I loved the chocolate, and was fine with the vanilla (wouldn’t usually choose it, but don’t disdain it).

That’s just my take on it: one flavor liked, one not liked, and one that’s just okay. Someone else might have the same pattern with different flavors. Or love them all equally, or want just the strawberry. Some might not like ice cream at all — any combination is possible.

What if we wanted to describe our feeling about Neapolitan as a whole?

Other ice cream flavors, Rocky Road, Rum Raisin, combine things and we treat them as a single flavor. Let’s do that with Neapolitan.

One consideration: there is a scale of how much we like a flavor: a little; so-so; a lot; not at all. Plus there are three distinct flavors, so it seems we need three distinct scales.

With most flavors we might be tempted to pick a number, say from one to ten, that says how much we like it. (Or one to one-hundred if we want really fine gradations. Or just one to three for simplicity. Whatever.)

We might use negative numbers if we want to include the notion of disliking something. Alternately, as with many surveys, we can use 1–10 with 5 defined as neutral and smaller numbers defined as negative feelings.[1]

None of this matters. All scales can be translated to all other scales. All that matters is that we have some notion of a “scale of liking.”

The important point is that there is a scale, and it offers a single number that represents our opinion on an ice cream flavor.


There seems a bit of a conundrum when it comes to Neapolitan ice cream.

It’s not hard to summarize our feelings about Banana Nut Fudge with a single value. Neapolitan, though, seems to divide us three ways.

We need three scales: one for chocolate, one for vanilla, one for strawberry. Our opinion on Neapolitan requires three numbers.

But this is treating them as three separate flavors; separate opinions about separate things, vanilla, chocolate, and strawberry.

We want to find a way to say the same thing with a single gesture, a way to say in one shot, “this is how I feel about Neapolitan ice cream.”


The Ice Cream Room (3D)

Vanilla: 5, Chocolate: 8, Strawberry: 3

Imagine a perfectly square room with a high ceiling, so the room is a big cube.

The one entrance is in the south-west corner. When you walk in, you’re given a round marker on a telescoping pole that can reach the ceiling.

From the south-west corner, your job is to:

  1. Walk straight east such that your distance reflects how much you like the vanilla in Neapolitan. Keep along the south wall as you do so. All the way to the east wall is total love for vanilla. If you totally hate it, stay put. If you’re neutral, walk halfway east.
  2. Now walk straight north to reflect how much you like chocolate. The same rules apply. All the way to the north wall is total love of chocolate. Staying where you are on the south wall means you hate it.
  3. Finally, plant your pole where you now stand, put the marker on it, and raise the pole to a height that reflects your love for strawberry. In this case the ceiling is total strawberry love, the floor is hating it.

You’ve set your marker in a location where the distance east represents your opinion of vanilla, the distance north your opinion of chocolate, and the height your opinion of strawberry.

All three opinions are encoded in the single location of the marker.

I want you to stop until you have a really good picture of this in your mind. The next few posts are all going to build on this image.


The Ice Cream Room (later that day)

More people placed markers. Most really like vanilla and chocolate.

Note that no one else’s marker would be in the exact same position unless they had the exact same three opinions about Neapolitan.

In the diagram directly above, seven people really like vanilla and chocolate but have varying love for strawberry. (And they all managed to vary slightly on the vanilla-chocolate love.)

If you needed a single number (from one to ten) to sum up your feelings, you might give a high number if you really liked all three flavors and a low number if you didn’t like ice cream.

But if you had mixed feelings you might give, say, a five. So might someone else with very different, but also mixed, feelings.

On a single scale, only very low and very high numbers tell us much. A “ten” is pretty clear. So is a “one.” But a “five” lacks information about what or why.

Our marker room provides a way to combine multiple opinions into a single marker without losing any information.

In the room, the equivalent of a “one” (ice cream hater) is the floor of the south-west corner where we started. Nothing east, nothing north, no height.

The equivalent of the “ten” is all the way to the north-east corner, and all the way to the ceiling.


Now imagine that many people have visited our Neapolitan room, and there are lots of markers, a forest of poles, all different heights:

The thicket of poles makes it hard to see the markers (which are all we care about), so from now on let’s imagine the poles are invisible:

Just always remember that the marker’s position is a combination of some amount east, some amount north, and some amount up.

That cloud of red markers represents the collective opinions of everyone who visited. Just looking at that cloud, it may be possible to see trends.


For instance, if very few poles are high, it means most people didn’t care for strawberry (which is certainly my view of the world 🙂 ).

If, in addition, the north part of the room is much denser than the south, it reflects a strong love of chocolate in the group.

Given that most people love ice cream, I would expect the south-west corner to be fairly sparsely populated. Likewise I’d expect the north-east corner to be heavily populated, reflecting a love of both vanilla and chocolate.

The Ice Cream Room

Lots of chocolate and vanilla lovers; not many on strawberry.


I hope you have a very clear picture in your mind of these markers that give us three distinct opinions in one.

What I’ve been describing is variously called a phase space, or a configuration space, or a parameter space (or vector space, but that comes later). The terms have specific meanings in specific fields, but they all share the same basic concept.[2]

The term configuration space is probably the most accessible: Our room reflects the configuration of an opinion on Neapolitan ice cream.

A collection of these opinions allows identification of trends or groupings in the space.

Further, the room represents the total possible space of any putative opinion on Neapolitan ice cream. Any opinion necessarily has a marker in the room, from haters at the door to lovers in the upper far corner.

The reverse is true: Every point in the space (room) represents an opinion about Neapolitan ice cream.


That’s all for this time.

I want to be sure you’ve got the image locked in. One location in the room marking three separate, unrelated opinions.

East for vanilla, North for chocolate, and Up for strawberry.

Next time I’ll talk about Baskin-Robbins.

And our room ain’t gonna work!

[1] I hate it when they do that. It just feels weird.

[2] I learned the concept as “phase space” and have used it for decades, so it’s kinda locked in. Be aware that, in physics, phase space refers to dynamical systems. What I’m discussing here is more akin to configuration or parameter space.

About Wyrd Smythe

The canonical fool on the hill watching the sunset and the rotation of the planet and thinking what he imagines are large thoughts. View all posts by Wyrd Smythe

15 responses to “3D Ice Cream

  • SelfAwarePatterns

    I have nothing mathy to contribute here. But I will note that having a hot brownie or couple of chocolate chips cookies under those ice cream scoops dramatically raises my opinion of the overall package!

  • Disagreeable Me (@Disagreeable_I)

    Seems like you’re gearing up for some larger point, which is fine.

    In the meantime, I’m wondering if you need to make some allowance for opinions where what matters is the combination. Like, I may quite like each individual flavour but love the combination. Butter OK. Toast is OK. Buttered toast is better than either.

    But this concern may not be relevant for your larger point.

    • Wyrd Smythe

      “Seems like you’re gearing up for some larger point, which is fine.”

      Yep. (Presenting configuration space as a very useful metaphor.)

      “Butter OK. Toast is OK. Buttered toast is better than either.”

      If we’re building a space for ‘things to eat’ butter probably isn’t an axis; it’s not something we normally eat. But one could certainly have a ‘dry toast’ dimension and a ‘buttered toast’ dimension if you cared about distinguishing those properties.

      You could, for instance, build a Toast Space with axes for degree of toasting, amount of butter (which would include zero), adding jam or jelly, using cream cheese, and so on.

      The space depends entirely on what you want to see!

      • Disagreeable Me (@Disagreeable_I)

        OK, so butter by itself is not a great option. But it is possible that there are people who prefer neapolitan ice cream to any one flavour, or indeed who dislike having these flavours mixed. For such people, a vector of how much they like each individual flavour does not really capture what they think of neapolitan ice cream.

        But if you’re just using it as an example of configuration space then this concern is not relevant, and I apologise for the digression!

      • Wyrd Smythe

        “But it is possible that there are people who prefer neapolitan ice cream to any one flavour, or indeed who dislike having these flavours mixed.”

        Absolutely! As you’ll see in the next post, if we wanted to distinguish Neapolitan as a flavor from other flavors of ice cream (including vanilla alone, chocolate alone, etc.) then we can create an “ice cream space” that allows us to compare Neapolitan to other ice creams.

        “For such people, a vector of how much they like each individual flavour does not really capture what they think of neapolitan ice cream.”

        Right. But the Neapolitan space is designed to describe how people feel about those three flavors. (I only started with Neapolitan because it’s a familiar thing that happens to have three equal components.) If we want to see other aspects of the Ice Cream Problem, we need to design other spaces.

        “I apologise for the digression!”

        No need to apologize! We’re way more on-topic than many comment discussions (most even?), and I’ve always seen the comment section as a way to further explore the post.

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